In line with the works in [1]}, [2]}, [3]}, our next goal is to design a Machine Learning method allowing us to directly obtain a solution of problem (REF ) for a parametric terminal condition \(g_K\) , for every value of the parameter \(K\in {\mathbb {R}}^p\)  in a compact set. In other words, we aim to learn the operator that maps the payoff function parameter \(K\)  to the solution of the PDE with terminal condition \(g_K\) .
We want to be able to train the neural networks once and for all on a selection of parameter values and obtain a network which takes a couple \((t,x)\)  and the parameter value \(K\)  and outputs the solution of the problem.
In [2]} the authors solve parametic PDEs by using a variant of highway networks [5]}, which are feedforward networks where each dense layer has an additional parameter called gate which allows the layer to output a combination of the unmodified input and of the output of the affine and activation operations, alleviating the vanishing gradient problem and allowing to train deeper networks. Their network takes the time, space and PDE parameter as input and is trained in the spirit of the Deep Galerkin method. The PDE residuals is computed from the neural network on a random time, space and parameter grid and is minimized in order to approximate the PDE solution. In [1]} the authors give four methods to solve parametric PDEs using a fully connected network taking the time, space and PDE parameter as input and minimizing a loss averaged over random parameter values.
In this article we follow the methodology developped in [3]} by relying on a class of neural networks, called DeepONet, presented in [8]}, and aiming to approximate functional operators.
This method is based on the following universal approximation theorem for operator, due to Chen and Chen [9]}.
